Electoral Manipulation under Media Influence (EMMI) models a setting in which an attacker selects a set of costly media strategies in order to persuade voters and make a designated candidate successful. Each strategy influences a subset of voters. A voter switches to the designated candidate once the number of selected strategies influencing that voter reaches a prescribed threshold. In this paper, we study the deterministic threshold model of EMMI under the plurality rule and the co-winner convention, focusing on the minimum-cost formulation.

We first obtain logarithmic inapproximability for Minimum-Cost EMMI, even for two candidates, unit strategy costs, and unit voter thresholds, and show that NP-hardness persists when every strategy influences only three voters. On the positive side, we formulate the unit-threshold case as an instance of Submodular Set Cover, obtaining a greedy $(1+\ln n)$-approximation for an arbitrary number of candidates, matching the logarithmic lower bound.

We further study the case where the influence sets are laminar. We prove that EMMI-Laminar is NP-hard when the number of candidates is part of the input, even under unit costs and unit thresholds. On the positive side, we give a polynomial-time algorithm for the two-candidate unit-cost unit-threshold laminar case. We then extend our algorithmic results to a fixed number of candidates, obtaining exact algorithms for the unit-cost unit-threshold case. We also give parameterized algorithms for unit-cost instances with arbitrary thresholds using chain decompositions. Finally, we present an integer linear programming formulation that provides an exact mathematical model for the problem.